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Pierre8rTeam
2018-03-09 16:43:19 +01:00
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//+------------------------------------------------------------------+
//| NoncentralF.mqh |
//| Copyright 2016-2017, MetaQuotes Software Corp. |
//| https://www.mql5.com |
//+------------------------------------------------------------------+
#property copyright "Copyright 2016-2017, MetaQuotes Software Corp."
#property link "https://www.mql5.com"
#include "Math.mqh"
#include "F.mqh"
#include "Gamma.mqh"
#include "NoncentralBeta.mqh"
//+------------------------------------------------------------------+
//| Noncentral-F probability density function (PDF) |
//+------------------------------------------------------------------+
//| The function returns the probability density function |
//| of the Noncentral-F distribution with parameters nu1,nu2,sigma. |
//| |
//| Arguments: |
//| x : Random variable |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| log_mode : Logarithm mode flag, if true it returns Log values |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The probability density evaluated at x. |
//+------------------------------------------------------------------+
double MathProbabilityDensityNoncentralF(const double x,const double nu1,const double nu2,const double sigma,const bool log_mode,int &error_code)
{
//--- return F if sigma==0
if(sigma==0.0)
return MathProbabilityDensityF(x,nu1,nu2,error_code);
//--- check NaN
if(!MathIsValidNumber(x) || !MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
{
error_code=ERR_ARGUMENTS_NAN;
return QNaN;
}
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
error_code=ERR_OK;
if(x<=0.0)
return TailLog0(true,log_mode);
//--- factors
double nu1_half=nu1*0.5;
double nu2_half=nu2*0.5;
double nu12_half=nu1_half+nu2_half;
double lambda=sigma*0.5;
double coef_lambda=MathExp(-lambda);
double nu_coef=nu1/nu2;
double g=x*nu_coef;
double pwr_g=MathExp((nu1_half-1)*MathLog(g));
double g1=g+1.0;
double pwr_g1=MathExp(-nu12_half*MathLog(g1));
double pwr_lambda=1.0;
double fact_mult=1.0;
//--- initial value for recurrent calculation
double r_beta=MathBeta(nu1_half,nu2_half);
//--- direct calculation of the sum
int max_terms=100;
int j=0;
double pdf=0;
while(j<max_terms)
{
if(j>0)
{
pwr_g*=g;
pwr_lambda*=lambda;
fact_mult/=j;
pwr_g1/=g1;
double jm1=j-1;
r_beta*=((nu1_half+jm1)/(nu12_half+jm1));
}
double dp=pwr_g*pwr_g1*coef_lambda*pwr_lambda*fact_mult/r_beta;
pdf+=dp;
if(dp/(pdf+10E-10)<10E-14)
break;
j++;
}
//--- check convergence
if(j<max_terms)
return TailLogValue(pdf*nu_coef,true,log_mode);
else
{
error_code=ERR_NON_CONVERGENCE;
return QNaN;
}
}
//+------------------------------------------------------------------+
//| Noncentral-F probability density function (PDF) |
//+------------------------------------------------------------------+
//| The function returns the probability density function |
//| of the Noncentral-F distribution with parameters nu1,nu2,sigma. |
//| |
//| Arguments: |
//| x : Random variable |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The probability density evaluated at x. |
//+------------------------------------------------------------------+
double MathProbabilityDensityNoncentralF(const double x,const double nu1,const double nu2,const double sigma,int &error_code)
{
return MathProbabilityDensityNoncentralF(x,nu1,nu2,sigma,false,error_code);
}
//+------------------------------------------------------------------+
//| Noncentral-F probability density function (PDF) |
//+------------------------------------------------------------------+
//| The function calculates the probability density function of |
//| the Noncentral F distribution with parameters nu1, nu2 and sigma |
//| for values in x[] array. |
//| |
//| Arguments: |
//| x : Array with random variables |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| log_mode : Logarithm mode flag, if true it returns Log values |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathProbabilityDensityNoncentralF(const double &x[],const double nu1,const double nu2,const double sigma,const bool log_mode,double &result[])
{
//--- return F if sigma==0
if(sigma==0.0)
return MathProbabilityDensityF(x,nu1,nu2,log_mode,result);
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
return false;
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
return false;
int data_count=ArraySize(x);
if(data_count==0)
return false;
const int max_terms=100;
//--- common factors
double nu1_half=nu1*0.5;
double nu2_half=nu2*0.5;
double nu12_half=nu1_half+nu2_half;
double lambda=sigma*0.5;
double coef_lambda=MathExp(-lambda);
double nu_coef=nu1/nu2;
double r_beta0=MathBeta(nu1_half,nu2_half);
ArrayResize(result,data_count);
for(int i=0; i<data_count; i++)
{
double x_arg=x[i];
if(x_arg<=0.0)
result[i]=TailLog0(true,log_mode);
else
{
double g=x_arg*nu_coef;
double g1=g+1.0;
//--- initial values for recurrent calculation
double pwr_g=MathExp((nu1_half-1)*MathLog(g));
double pwr_g1=MathExp(-nu12_half*MathLog(g1));
double pwr_lambda=1.0;
double fact_mult=1.0;
double r_beta=r_beta0;
//--- direct calculation of the sum
int j=0;
double pdf=0;
while(j<max_terms)
{
if(j>0)
{
pwr_g*=g;
pwr_lambda*=lambda;
fact_mult/=j;
pwr_g1/=g1;
double jm1=j-1;
r_beta*=((nu1_half+jm1)/(nu12_half+jm1));
}
double dp=pwr_g*pwr_g1*coef_lambda*pwr_lambda*fact_mult/r_beta;
pdf+=dp;
if(dp/(pdf+10E-10)<10E-14)
break;
j++;
}
//--- check convergence
if(j<max_terms)
result[i]=TailLogValue(pdf*nu_coef,true,log_mode);
else
return false;
}
}
return true;
}
//+------------------------------------------------------------------+
//| Noncentral-F probability density function (PDF) |
//+------------------------------------------------------------------+
//| The function calculates the probability density function of |
//| the Noncentral F distribution with parameters nu1, nu2 and sigma |
//| for values in x[] array. |
//| |
//| Arguments: |
//| x : Array with random variables |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathProbabilityDensityNoncentralF(const double &x[],const double nu1,const double nu2,const double sigma,double &result[])
{
return MathProbabilityDensityNoncentralF(x,nu1,nu2,sigma,false,result);
}
//+------------------------------------------------------------------+
//| Noncentral F cumulative distribution function (CDF) |
//+------------------------------------------------------------------+
//| The function returns the probability that an observation |
//| from Noncentral F distribution with parameters nu1,nu2,sigma |
//| is less than or equal to x. |
//| |
//| Arguments: |
//| x : The desired quantile |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| tail : Flag to calculate lower tail |
//| log_mode : Logarithm mode, if true it calculates Log values |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The value of the Noncentral F cumulative distribution function |
//| with parameters nu1,nu2,sigma, evaluated at x. |
//+------------------------------------------------------------------+
double MathCumulativeDistributionNoncentralF(const double x,const double nu1,const double nu2,const double sigma,const bool tail,const bool log_mode,int &error_code)
{
//--- return F if sigma==0
if(sigma==0.0)
return MathCumulativeDistributionF(x,nu1,nu2,error_code);
//--- check NaN
if(!MathIsValidNumber(x) || !MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
{
error_code=ERR_ARGUMENTS_NAN;
return QNaN;
}
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0 || x<0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
error_code=ERR_OK;
if(x<=0)
return TailLog0(tail,log_mode);
//--- calculate cdf using Noncentral Beta distribution
double arg=(nu1/nu2)*x;
return MathCumulativeDistributionNoncentralBeta(arg/(1.0+arg),nu1*0.5,nu2*0.5,sigma,tail,log_mode,error_code);
}
//+------------------------------------------------------------------+
//| Noncentral F cumulative distribution function (CDF) |
//+------------------------------------------------------------------+
//| The function returns the probability that an observation |
//| from Noncentral F distribution with parameters nu1,nu2,sigma |
//| is less than or equal to x. |
//| |
//| Arguments: |
//| x : The desired quantile |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The value of the Noncentral F cumulative distribution function |
//| with parameters nu1,nu2,sigma, evaluated at x. |
//+------------------------------------------------------------------+
double MathCumulativeDistributionNoncentralF(const double x,const double nu1,const double nu2,const double sigma,int &error_code)
{
return MathCumulativeDistributionNoncentralF(x,nu1,nu2,sigma,true,false,error_code);
}
//+------------------------------------------------------------------+
//| Noncentral F cumulative distribution function (CDF) |
//+------------------------------------------------------------------+
//| The function calculates the cumulative distribution function of |
//| the Noncentral Fl distribution with parameters nu1,nu2 and sigma |
//| for values in x[] array. |
//| |
//| Arguments: |
//| x : Array with random variables |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| tail : Flag to calculate lower tail |
//| log_mode : Logarithm mode, if true it calculates Log values |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathCumulativeDistributionNoncentralF(const double &x[],const double nu1,const double nu2,const double sigma,const bool tail,const bool log_mode,double &result[])
{
//--- return F if sigma==0
if(sigma==0.0)
return MathCumulativeDistributionF(x,nu1,nu2,tail,log_mode,result);
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
return false;
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
return false;
int data_count=ArraySize(x);
if(data_count==0)
return false;
//--- common constants
int error_code=0;
double nu1_half=nu1*0.5;
double nu2_half=nu2*0.5;
ArrayResize(result,data_count);
for(int i=0; i<data_count; i++)
{
double x_arg=x[i];
if(x_arg<=0)
result[i]=TailLog0(tail,log_mode);
else
{
//--- calculate cdf using Noncentral Beta distribution
double arg=(nu1/nu2)*x_arg;
result[i]=MathCumulativeDistributionNoncentralBeta(arg/(1.0+arg),nu1_half,nu2_half,sigma,tail,log_mode,error_code);
//--- check result
if(error_code!=ERR_OK)
return false;
}
}
return true;
}
//+------------------------------------------------------------------+
//| Noncentral F cumulative distribution function (CDF) |
//+------------------------------------------------------------------+
//| The function calculates the cumulative distribution function of |
//| the Noncentral Fl distribution with parameters nu1,nu2 and sigma |
//| for values in x. |
//| Arguments: |
//| x : Array with random variables |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathCumulativeDistributionNoncentralF(const double &x[],const double nu1,const double nu2,const double sigma,double &result[])
{
return MathCumulativeDistributionNoncentralF(x,nu1,nu2,sigma,true,false,result);
}
//+------------------------------------------------------------------+
//| Noncentral F distribution quantile function (inverse CDF) |
//+------------------------------------------------------------------+
//| The function returns the inverse cumulative distribution |
//| function of Noncentral F distribution with parameters nu1,nu2 |
//| and sigma for the desired probability. |
//| |
//| Arguments: |
//| probability : The desired probability |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| tail : Flag to calculate lower tail |
//| log_mode : Logarithm mode, if true it calculates Log values |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The value of the inverse Noncentral F cumulative distribution |
//| function with parameters nu1,nu2,sigma, evaluated at x. |
//+------------------------------------------------------------------+
double MathQuantileNoncentralF(const double probability,const double nu1,const double nu2,const double sigma,const bool tail,const bool log_mode,int &error_code)
{
if(log_mode==true && probability==QNEGINF)
return 0.0;
//--- return F if sigma==0
if(sigma==0.0)
return MathQuantileF(probability,nu1,nu2,tail,log_mode,error_code);
//--- check NaN
if(!MathIsValidNumber(probability) || !MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
{
error_code=ERR_ARGUMENTS_NAN;
return QNaN;
}
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
//--- check sigma
if(sigma<0.0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
//--- calculate real probability
double prob=TailLogProbability(probability,tail,log_mode);
//--- check probability range
if(prob<0.0 || prob>1.0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
if(prob==1.0)
{
error_code=ERR_RESULT_INFINITE;
return QPOSINF;
}
error_code=ERR_OK;
if(prob==0.0)
return 0.0;
//---
int max_iterations=50;
int iterations=0;
//--- initial values
double h=1.0;
double h_min=10E-10;
double x=0.5;
int err_code=0;
//--- Newton iterations
while(iterations<max_iterations)
{
//--- check convegence
if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
break;
//--- calculate pdf and cdf
double pdf=MathProbabilityDensityNoncentralF(x,nu1,nu2,sigma,err_code);
double cdf=MathCumulativeDistributionNoncentralF(x,nu1,nu2,sigma,err_code);
//--- calculate ratio
h=(cdf-prob)/pdf;
//---
double x_new=x-h;
//--- check x
if(x_new<0.0)
x_new=x*0.1;
else
if(x_new>1.0)
x_new=1.0-(1.0-x)*0.1;
x=x_new;
iterations++;
}
//--- check convergence
if(iterations<max_iterations)
return x;
else
{
error_code=ERR_NON_CONVERGENCE;
return QNaN;
}
}
//+------------------------------------------------------------------+
//| Noncentral F distribution quantile function (inverse CDF) |
//+------------------------------------------------------------------+
//| The function returns the inverse cumulative distribution |
//| function of Noncentral F distribution with parameters nu1,nu2 |
//| and sigma for the desired probability. |
//| |
//| Arguments: |
//| probability : The desired probability |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The value of the inverse Noncentral F cumulative distribution |
//| function with parameters nu1,nu2,sigma, evaluated at x. |
//+------------------------------------------------------------------+
double MathQuantileNoncentralF(const double probability,const double nu1,const double nu2,const double sigma,int &error_code)
{
return MathQuantileNoncentralF(probability,nu1,nu2,sigma,true,false,error_code);
}
//+------------------------------------------------------------------+
//| Noncentral F distribution quantile function (inverse CDF) |
//+------------------------------------------------------------------+
//| The function returns the inverse cumulative distribution |
//| function of Noncentral F distribution with parameters nu1,nu2 |
//| and sigma for values from the probability[] array. |
//| |
//| Arguments: |
//| probability : Array with probabilities |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| tail : Flag to calculate lower tail |
//| log_mode : Logarithm mode, if true it calculates Log values |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathQuantileNoncentralF(const double &probability[],const double nu1,const double nu2,const double sigma,const bool tail,const bool log_mode,double &result[])
{
//--- return F if sigma==0
if(sigma==0.0)
return MathQuantileF(probability,nu1,nu2,tail,log_mode,result);
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
return false;
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
return false;
//--- check sigma
if(sigma<0.0)
return false;
int data_count=ArraySize(probability);
if(data_count==0)
return false;
int error_code=0;
ArrayResize(result,data_count);
for(int i=0; i<data_count; i++)
{
//--- calculate real probability
double prob=TailLogProbability(probability[i],tail,log_mode);
//--- check probability range
if(prob<0.0 || prob>1.0)
return false;
if(prob==1.0)
result[i]=QPOSINF;
else
if(prob==0.0)
result[i]=0.0;
else
{
int max_iterations=50;
int iterations=0;
//--- initial values
double h=1.0;
double h_min=10E-10;
double x=0.5;
int err_code=0;
//--- Newton iterations
while(iterations<max_iterations)
{
//--- check convegence
if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
break;
//--- calculate pdf and cdf
double pdf=MathProbabilityDensityNoncentralF(x,nu1,nu2,sigma,err_code);
double cdf=MathCumulativeDistributionNoncentralF(x,nu1,nu2,sigma,err_code);
//--- calculate ratio
h=(cdf-prob)/pdf;
//---
double x_new=x-h;
//--- check x
if(x_new<0.0)
x_new=x*0.1;
else
if(x_new>1.0)
x_new=1.0-(1.0-x)*0.1;
x=x_new;
iterations++;
}
//--- check convergence
if(iterations<max_iterations)
result[i]=x;
else
return false;
}
}
return true;
}
//+------------------------------------------------------------------+
//| Noncentral F distribution quantile function (inverse CDF) |
//+------------------------------------------------------------------+
//| The function returns the inverse cumulative distribution |
//| function of Noncentral F distribution with parameters nu1,nu2 |
//| and sigma for values from the probability[] array. |
//| |
//| Arguments: |
//| probability : Array with probabilities |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| result : Array with calculated values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathQuantileNoncentralF(const double &probability[],const double nu1,const double nu2,const double sigma,double &result[])
{
return MathQuantileNoncentralF(probability,nu1,nu2,sigma,true,false,result);
}
//+------------------------------------------------------------------+
//| Random variate from the Noncentral F-distribution |
//+------------------------------------------------------------------+
//| Compute the random variable from the Noncentral F-distribution |
//| with parameters nu1, nu2 and sigma. |
//| |
//| Arguments: |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| The random value with Noncentral F-distribution. |
//+------------------------------------------------------------------+
double MathRandomNoncentralF(const double nu1,const double nu2,const double sigma,int &error_code)
{
//--- return F if sigma==0
if(sigma==0.0)
return MathRandomF(nu1,nu2,error_code);
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
{
error_code=ERR_ARGUMENTS_NAN;
return QNaN;
}
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
//--- check sigma
if(sigma<0.0)
{
error_code=ERR_ARGUMENTS_INVALID;
return QNaN;
}
error_code=ERR_OK;
//--- calculate using noncentral chisquare and chisquare distributions
double num=MathRandomNoncentralChiSquare(nu1,sigma,error_code)*nu2;
double den=MathRandomGamma(nu2*0.5,2.0,error_code)*nu1;
if(den!=0)
return num/den;
else
{
error_code=ERR_NON_CONVERGENCE;
return QNaN;
}
}
//+------------------------------------------------------------------+
//| Random variate from the Noncentral F distribution |
//+------------------------------------------------------------------+
//| Generates random variables from the Noncentral F distribution |
//| with parameters nu1, nu2 and sigma. |
//| |
//| Arguments: |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| data_count : Number of values needed |
//| result : Output array with random values |
//| |
//| Return value: |
//| true if successful, otherwise false. |
//+------------------------------------------------------------------+
bool MathRandomNoncentralF(const double nu1,const double nu2,const double sigma,const int data_count,double &result[])
{
//--- return F if sigma==0
if(sigma==0.0)
return MathRandomF(nu1,nu2,data_count,result);
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
return false;
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
return false;
//--- check sigma
if(sigma<0.0)
return false;
int error_code=0;
//--- prepare output array and calculate random values
ArrayResize(result,data_count);
for(int i=0; i<data_count; i++)
{
//--- calculate using noncentral chisquare and chisquare distributions
double num=MathRandomNoncentralChiSquare(nu1,sigma,error_code)*nu2;
double den=MathRandomGamma(nu2*0.5,2.0,error_code)*nu1;
if(den!=0)
result[i]=num/den;
else
return false;
}
return true;
}
//+------------------------------------------------------------------+
//| Noncentral F distribution moments |
//+------------------------------------------------------------------+
//| The function calculates 4 first moments of the Noncental F |
//| distribution with parameters nu1,nu2 and sigma. |
//| |
//| Arguments: |
//| nu1 : Numerator degrees of freedom |
//| nu2 : Denominator degrees of freedom |
//| sigma : Noncentrality parameter |
//| mean : Variable for mean value (1st moment) |
//| variance : Variable for variance value (2nd moment) |
//| skewness : Variable for skewness value (3rd moment) |
//| kurtosis : Variable for kurtosis value (4th moment) |
//| error_code : Variable for error code |
//| |
//| Return value: |
//| true if moments calculated successfully, otherwise false. |
//+------------------------------------------------------------------+
bool MathMomentsNoncentralF(const double nu1,const double nu2,const double sigma,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code)
{
//--- if sigma==0, calc moments for F
if(sigma==0)
return MathMomentsF(nu1,nu2,mean,variance,skewness,kurtosis,error_code);
//--- default values
mean =QNaN;
variance=QNaN;
skewness=QNaN;
kurtosis=QNaN;
//--- check NaN
if(!MathIsValidNumber(nu1) || !MathIsValidNumber(nu2) || !MathIsValidNumber(sigma))
{
error_code=ERR_ARGUMENTS_NAN;
return false;
}
//--- check arguments
if(nu1!=MathRound(nu1) || nu2!=MathRound(nu2) || nu1<=0 || nu2<=0)
{
error_code=ERR_ARGUMENTS_INVALID;
return false;
}
//--- check sigma
if(sigma<0.0)
{
error_code=ERR_ARGUMENTS_INVALID;
return false;
}
error_code=ERR_OK;
//--- calculate moments
if(nu2>2)
mean=nu2*(nu1+sigma)/(nu1*(nu2-2));
//--- variance
if(nu2>4)
variance=2*MathPow(nu2/nu1,2)*((nu2-2)*(nu1+2*sigma)+MathPow(nu1+sigma,2))/((nu2-4)*MathPow(nu2-2,2));
//--- factors
double sigma_sqr=MathPow(sigma,2);
double sigma_cube=sigma_sqr*sigma;
double nu12m2=(nu1+nu2-2);
double nu2p10=(nu2+10);
//--- skewness
if(nu2>6)
{
skewness=2*M_SQRT2*MathSqrt(nu2-4);
skewness*=(nu12m2*(6*sigma_sqr+(2*nu1+nu2-2)*(3*sigma+nu1))+2*sigma_cube);
skewness/=(nu2-6);
skewness/=MathPow(nu12m2*(2*sigma+nu1)+sigma_sqr,1.5);
}
//--- kurtosis
if(nu2>8)
{
double coef=nu2p10*(MathPow(nu1,2)+nu1*(nu2-2))+4*MathPow(nu2-2,2);
kurtosis=1;
kurtosis=3*(nu2-4);
kurtosis*=(nu12m2*(coef*(4*sigma+nu1)+nu2p10*(4*sigma_cube+2*sigma_sqr*(3*nu1+2*nu2-4)))+nu2p10*MathPow(sigma,4));
kurtosis/=(nu2-8)*(nu2-6);
kurtosis/=MathPow((nu12m2*(2*sigma+nu1)+sigma_sqr),2);
kurtosis-=3;
}
//--- successful
return true;
}
//+------------------------------------------------------------------+